Learn Turkish: Ben anaokulunda çok ağlamışım. - Apparently I cried a lot at kindergarten.

Anaokulu (kindergarten) is a compound word made of ana (main) and okul (school). Because it is a compound noun, it already has a possessive suffix at the end (-u). Whenever you add a location ending like -da (at/in) to a compound noun, you must use the buffer letter -n- instead of the usual -y-. So it becomes anaokulunda.

Why does the verb use "-mış" (ağlamışım) instead of the regular past tense "-dı" (ağladım)?

The -mış suffix is the reported or evidential past tense. You use it when you didn't witness an event or don't personally remember it, but heard about it from someone else. Since the speaker is talking about when they were a toddler in kindergarten, they probably don't actually remember crying, but their parents or teachers told them they did!

Why does ağlamışım end with "-ım" instead of just "-m" like in ağladım?

Even though -mış is a past tense, it pairs with the present tense personal endings (the "to be" suffixes), not the special endings used for the regular -dı past tense. Therefore, for the "I" subject, you attach -ım to get ağlamışım.

Could I just say "Anaokulunda çok ağlamışım" without the word "Ben"?

Yes! The -ım ending on ağlamışım already tells us "I" am the subject. Adding Ben at the beginning just gives a little extra emphasis to the sentence, but it is completely optional.